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Concept

Resolution

A proof rule that combines two clauses disagreeing about one variable into a single clause that leaves that variable out. Repeated until the empty clause appears, it proves that a set of clauses cannot all be true, and it is the reasoning inside modern satisfiability solvers.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

A resolution refutation of four clauses on three variables. A derivation tree: the given clauses at the top, each later clause obtained by cancelling one variable between two clauses above it, ending in the empty clause.

A proof with one rule

Two clauses that disagree about exactly one variable can be combined into a third that forgets it; repeat, and if the clauses cannot all be true the empty clause eventually appears — a complete proof system with a single move.

logic · Resolution 5 two-literal clauses drawn as 10 arrows. The literals of four variables arranged in a circle with an arrow for each implication a two-literal clause contains, the arrows forming a path from one literal to its negation and back highlighted, beside the list of clauses and their arrows.

Two literals make an arrow

A clause of two literals, p ∨ q, says that if p is false then q is true, and if q is false then p is true — two arrows. A set of such clauses is a directed graph on the literals, and it is unsatisfiable exactly when some variable and its negation reach each other. Each of those two paths is a chain of resolution steps, and finding them takes time proportional to the size of the input.

logic · Resolution A failed search on four clauses, read as a resolution refutation. A binary search tree branching on variables, each branch ending at a clause the partial assignment makes false, with every branch point labelled by the resolvent of the clauses below it, the top label being the empty clause.

A failed search is a proof

Search for an assignment by branching on variables and backing up whenever a clause turns false. If every branch fails, the tree the search leaves behind is itself a resolution refutation: write at each branch point the resolvent of the clauses below it, and the top of the tree is the empty clause. So every limit on short refutations is a limit on every such search — and the pigeonhole clauses, whose refutations are long, defeat them all.

logic · Resolution

Named alongside it

The objects these essays reach for when they reach for this one.

RefutationSatisfiabilityLiteralNormal formProof systemCompletenessComplexityExhaustive searchGraphPigeonhole principleSoundnessThreshold

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